Black-Scholes-Merton Model

As Δt → 0, the number of nodes in our binomial lattice goes to infinity and we are essentially moving to a continuous time option model. The Black-Scholes model (independently derived by Robert Merton) is a closed form continuous time option pricing model. Its derivation is mathematically rigorous, but I think we can sketch its derivation using the tools we've derived thus far. The model is based on the assumptions that the underlying stock price is a geometric Brownian motion process and that the stock price S has a lognormal distribution. The intuition is this: there are three assets—a bond, a stock, and a derivative. These assets’ dynamics can be described by the following respective models, which we've already derived:

We recognize, first, the deterministic return on a riskless bond followed by geometric Brownian motion describing the stock price movement and, finally, Ito's lemma, which describes the dynamics underlying the derivative on the stock. The insight behind Black-Scholes is that we could form a riskless portfolio consisting of the stock and the derivative, thus equating the return on this portfolio to the return on the riskless bond. This is the notion of constructing a replicating portfolio. Suppose we do this, selecting X units of the stock and Y units of the derivative. Then, at time t, we have the following portfolio:

To replicate the payout of a bond, we choose X and Y so that this portfolio is riskless. Since it is riskless, this implies that it earns the same return as the bond, which means that . This condition must be satisfied, otherwise there would be an arbitrage opportunity suggesting that one could short the security with the lower return and go long the other, earning a riskless premium. We need an expression for . To get this, we use the simple definition:

As long as no money is added or taken from this portfolio, it will be self-financing and therefore have the following dynamic:

We can substitute for dS and dF to get:

Collecting terms:

This simplifies. The reason is that we need this portfolio to be riskless. For that to happen, we must first eliminate dz as follows:

This implies that , implying that . This will allow us to eliminate X as well. Making these substitutions, we get:

But,

Therefore,

Finally, eliminating Y and dt solves for the Black-Scholes equation:

This is a partial differential equation. Its solution depends on the option and the option's boundary condition. For European call options, we impose the boundary condition that c(0,t) = 0 (where c is the call option) and we want the value of the call to maximize the (risk neutral) expected value of the future stock price over the strike price, for example,

The solution to the partial differential for this boundary condition is:

with d₁ and d₂ given by:

N(d₁) and N(d₂) are cumulative standard normal densities (NORMSDIST in Excel). Admittedly, this looks intimidating. If you look at the call option value, the term in brackets consists of the difference between the expected growth in the stock price over the strike price K. The normal densities essentially are derived from the expected value of a lognormally distributed random variable (the stock price). This difference is discounted using the risk-free rate over the period of the option. Simplifying, we get:

Again, we see that the value of the call is the difference between today's stock price and the discounted exercise price, both scaled by a constant evaluated under the cumulative standard normal distribution. The parameters of d₁ and d₂ clearly reflect the properties of the lognormal density, for example, (), σ√T.